Some Global Aspects of Gauge Anomalies of Semisimple Gauge Groups and Fermion Generations in Gut and Superstring Theories

Abstract

We study more extensively and completely for global gauge anomalies with some semisimple gauge groups as initiated in ref.1. A detailed and complete proof or derivation is provided for the Z 2 global (nonperturbative) gauge anomaly given in ref.1 for a gauge theory with the semisimple gauge group SU(2) × SU(2) × SU(2) in D = 4 dimensions and Weyl fermions in the irreducible representation (IR) ω = (2, 2, 2) with 2 denoting the corresponding dimensions. This Z 2 anomaly was used in the discussions related to generic SO(10) and supersymmetric SO(10) unification theories 1 for the total generation numbers of fermions and mirror fermions. Our result 1 that the global anomaly coefficient formula is given by A(ω) = exp[iπQ 2 ( )] = −1 in this case with Q 2 ( ) being the Dynkin index for SU(8) in the fundamental IR ( ) = (8) and that the corresponding gauge transformations need to be topologically non-trivial simultaneously in all the three SU(2) factors for the homotopy group Π 4 (SU(2) × SU(2) × SU(2)) is also discussed, and as shown by our results 1 the semisimple gauge transformations collectively may have physical consequences which do not correspond to successive simple gauge transformations. The similar result given in ref.1 for the Z 2 global gauge anomaly of gauge group SU(2) × SU(2) with Weyl fermions in the IR ω = (2, 2) with 2 denoting the corresponding dimensions is also discussed with proof similar to the case of SU(2) × SU(2) × SU(2). We also give a complete proof for some relevent topological results. We expect that our results and discussions may also be useful in more general studies related to global aspects of gauge theories. Gauge anomalies for the relevant semisimple gauge groups are also briefly discussed in higher dimensions, especially for self-contragredient representations, with discussions involving trace identities relating to ref.14. We also relate the discussions to our results and propositions in our previous studies of global gauge anomalies. We also remark the connection of our results and discussions to the total generation numbers in relevant theories.